wave, ocean wave

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 Wave From Wikipedia, the free encyclopedia This article is about waves in the scientific sense. For waves the surface of the ocean or lakes, see Wind wave. For other uses, see  Wave (disambiguation) . Surface waves  in water In  physics, a wave is disturbance or oscillation (of a physical quantity), that travels through matter or space, accompanied by a transfer of energy. Wave motion transfers  energy  from one  point to another, often with no permanent displacement of the par ticles of the medium   that is, with little or no associated mass transport. They consist, instead, of  oscillations or vibrations around almost fixed locations. Waves are described by a  wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave. There are two main types of waves.  Mechanical waves   propagate through a medium, and the substance of this medium is deformed. The deformation reverses itself owing to  restoring forces  resulting from its deformation. For example, sound waves propagate via air molecules colliding with their neighbors. When air molecules collide, they also bounce away from each other (a restoring force). This keeps the molecules from continuing to travel in the direction of the wave. The second main type of wave,  electromagnetic waves, do not require a medium. Instead, they consist of periodic oscillations of electrical and magnetic fields generated by charged particles, and can therefore travel through a  vacuum. These types of waves vary in wavelength, and include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays. Further, the behavior of particles in quantum mechanics are described by waves. In addition, gravitational waves also travel through space, which are a result of a vibration or movement in gravitational fields. A wave can be transverse or  longitudinal depending on the direction of its oscillation. Transverse waves occur when a disturbance creates oscillations that are perpendicular (at right angles) to the  propagation (the direction of energy transf er). Longitudinal waves occur when the oscillations

description

In physics, a wave is disturbance or oscillation (of a physical quantity), that travels through matter or space, accompanied by a transfer of energy. Wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations around almost fixed locations. Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematical form of this equation varies depending on the type of wave.

Transcript of wave, ocean wave

  • Wave

    From Wikipedia, the free encyclopedia

    This article is about waves in the scientific sense. For waves the surface of the ocean or lakes,

    see Wind wave. For other uses, see Wave (disambiguation).

    Surface waves in water

    In physics, a wave is disturbance or oscillation (of a physical quantity), that travels through

    matter or space, accompanied by a transfer of energy. Wave motion transfers energy from one

    point to another, often with no permanent displacement of the particles of the mediumthat is, with little or no associated mass transport. They consist, instead, of oscillations or vibrations

    around almost fixed locations. Waves are described by a wave equation which sets out how the

    disturbance proceeds over time. The mathematical form of this equation varies depending on the

    type of wave.

    There are two main types of waves. Mechanical waves propagate through a medium, and the

    substance of this medium is deformed. The deformation reverses itself owing to restoring forces

    resulting from its deformation. For example, sound waves propagate via air molecules colliding

    with their neighbors. When air molecules collide, they also bounce away from each other (a

    restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

    The second main type of wave, electromagnetic waves, do not require a medium. Instead, they

    consist of periodic oscillations of electrical and magnetic fields generated by charged particles,

    and can therefore travel through a vacuum. These types of waves vary in wavelength, and

    include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays,

    and gamma rays.

    Further, the behavior of particles in quantum mechanics are described by waves. In addition,

    gravitational waves also travel through space, which are a result of a vibration or movement in

    gravitational fields.

    A wave can be transverse or longitudinal depending on the direction of its oscillation. Transverse

    waves occur when a disturbance creates oscillations that are perpendicular (at right angles) to the

    propagation (the direction of energy transfer). Longitudinal waves occur when the oscillations

  • are parallel to the direction of propagation. While mechanical waves can be both transverse and

    longitudinal, all electromagnetic waves are transverse in free space.

    Contents

    1 General features

    2 Mathematical description of one-dimensional waves

    o 2.1 Wave equation

    o 2.2 Wave forms

    o 2.3 Amplitude and modulation

    o 2.4 Phase velocity and group velocity

    3 Sinusoidal waves

    4 Plane waves

    5 Standing waves

    6 Physical properties

    o 6.1 Transmission and media

    o 6.2 Absorption

    o 6.3 Reflection

    o 6.4 Interference

    o 6.5 Refraction

    o 6.6 Diffraction

    o 6.7 Polarization

    o 6.8 Dispersion

    7 Mechanical waves

    o 7.1 Waves on strings

    o 7.2 Acoustic waves

    o 7.3 Water waves

    o 7.4 Seismic waves

    o 7.5 Shock waves

    o 7.6 Other

    8 Electromagnetic waves

    9 Quantum mechanical waves

    o 9.1 de Broglie waves

    10 Gravity waves

    11 Gravitational waves

    12 WKB method

    13 See also

    14 References

    15 Sources

    16 External links

    General features

    A single, all-encompassing definition for the term wave is not straightforward. A vibration can

    be defined as a back-and-forth motion around a reference value. However, a vibration is not

  • necessarily a wave. An attempt to define the necessary and sufficient characteristics that qualify

    a phenomenon to be called a wave results in a fuzzy border line.

    The term wave is often intuitively understood as referring to a transport of spatial disturbances

    that are generally not accompanied by a motion of the medium occupying this space as a whole.

    In a wave, the energy of a vibration is moving away from the source in the form of a disturbance

    within the surrounding medium (Hall 1980, p. 8). However, this notion is problematic for a

    standing wave (for example, a wave on a string), where energy is moving in both directions

    equally, or for electromagnetic (e.g., light) waves in a vacuum, where the concept of medium

    does not apply and interaction with a target is the key to wave detection and practical

    applications. There are water waves on the ocean surface; gamma waves and light waves emitted

    by the Sun; microwaves used in microwave ovens and in radar equipment; radio waves broadcast

    by radio stations; and sound waves generated by radio receivers, telephone handsets and living

    creatures (as voices), to mention only a few wave phenomena.

    It may appear that the description of waves is closely related to their physical origin for each

    specific instance of a wave process. For example, acoustics is distinguished from optics in that

    sound waves are related to a mechanical rather than an electromagnetic wave transfer caused by

    vibration. Concepts such as mass, momentum, inertia, or elasticity, become therefore crucial in

    describing acoustic (as distinct from optic) wave processes. This difference in origin introduces

    certain wave characteristics particular to the properties of the medium involved. For example, in

    the case of air: vortices, radiation pressure, shock waves etc.; in the case of solids: Rayleigh

    waves, dispersion; and so on.

    Other properties, however, although usually described in terms of origin, may be generalized to

    all waves. For such reasons, wave theory represents a particular branch of physics that is

    concerned with the properties of wave processes independently of their physical origin.[1]

    For

    example, based on the mechanical origin of acoustic waves, a moving disturbance in spacetime can exist if and only if the medium involved is neither infinitely stiff nor infinitely pliable. If all

    the parts making up a medium were rigidly bound, then they would all vibrate as one, with no

    delay in the transmission of the vibration and therefore no wave motion. On the other hand, if all

    the parts were independent, then there would not be any transmission of the vibration and again,

    no wave motion. Although the above statements are meaningless in the case of waves that do not

    require a medium, they reveal a characteristic that is relevant to all waves regardless of origin:

    within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different

    for adjacent points in space because the vibration reaches these points at different times.

    Mathematical description of one-dimensional waves

    Wave equation

    Main articles: Wave equation and D'Alembert's formula

    Consider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider

    the string to have a single spatial dimension. Consider this wave as traveling

  • Wavelength , can be measured between any two corresponding points on a waveform

    in the direction in space. E.g., let the positive direction be to the right, and the

    negative direction be to the left.

    with constant amplitude

    with constant velocity , where is

    o independent of wavelength (no dispersion)

    o independent of amplitude (linear media, not nonlinear).[2]

    with constant waveform, or shape

    This wave can then be described by the two-dimensional functions

    (waveform traveling to the right)

    (waveform traveling to the left)

    or, more generally, by d'Alembert's formula:[3]

    representing two component waveforms and traveling through the medium in opposite

    directions. A generalized representation of this wave can be obtained[4]

    as the partial differential

    equation

    General solutions are based upon Duhamel's principle.[5]

    Wave forms

  • Sine, square, triangle and sawtooth waveforms.

    The form or shape of F in d'Alembert's formula involves the argument x vt. Constant values of this argument correspond to constant values of F, and these constant values occur if x increases

    at the same rate that vt increases. That is, the wave shaped like the function F will move in the

    positive x-direction at velocity v (and G will propagate at the same speed in the negative x-

    direction).[6]

    In the case of a periodic function F with period , that is, F(x + vt) = F(x vt), the periodicity of F in space means that a snapshot of the wave at a given time t finds the wave varying

    periodically in space with period (the wavelength of the wave). In a similar fashion, this periodicity of F implies a periodicity in time as well: F(x v(t + T)) = F(x vt) provided vT = , so an observation of the wave at a fixed location x finds the wave undulating periodically in time

    with period T = /v.[7]

    Amplitude and modulation

    Illustration of the envelope (the slowly varying red curve) of an amplitude-modulated wave. The

    fast varying blue curve is the carrier wave, which is being modulated.

    Main article: Amplitude modulation

    See also: Frequency modulation and Phase modulation

    The amplitude of a wave may be constant (in which case the wave is a c.w. or continuous wave),

    or may be modulated so as to vary with time and/or position. The outline of the variation in

  • amplitude is called the envelope of the wave. Mathematically, the modulated wave can be written

    in the form:[8][9][10]

    where is the amplitude envelope of the wave, is the wavenumber and is the phase. If

    the group velocity (see below) is wavelength-independent, this equation can be simplified

    as:[11]

    showing that the envelope moves with the group velocity and retains its shape. Otherwise, in

    cases where the group velocity varies with wavelength, the pulse shape changes in a manner

    often described using an envelope equation.[11][12]

    Phase velocity and group velocity

    Main articles: Phase velocity and Group velocity

    See also: Envelope (waves) Phase and group velocity

    Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot

    moves with the phase velocity, and the green dots propagate with the group velocity.

    There are two velocities that are associated with waves, the phase velocity and the group

    velocity. To understand them, one must consider several types of waveform. For simplification,

    examination is restricted to one dimension.

    This shows a wave with the Group velocity and Phase velocity going in different directions.

    The most basic wave (a form of plane wave) may be expressed in the form:

    which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the

    argument, , makes clear that this expression describes a

  • vibration of wavelength traveling in the x-direction with a constant phase velocity

    .[13]

    The other type of wave to be considered is one with localized structure described by an envelope,

    which may be expressed mathematically as, for example:

    where now A(k1) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a

    sharp peak in a region of wave vectors k surrounding the point k1 = k. In exponential form:

    with Ao the magnitude of A. For example, a common choice for Ao is a Gaussian wave packet:[14]

    where determines the spread of k1-values about k, and N is the amplitude of the wave.

    The exponential function inside the integral for oscillates rapidly with its argument, say (k1), and where it varies rapidly, the exponentials cancel each other out, interfere destructively,

    contributing little to .[13] However, an exception occurs at the location where the argument of the exponential varies slowly. (This observation is the basis for the method of stationary phase

    for evaluation of such integrals.[15]

    ) The condition for to vary slowly is that its rate of change with k1 be small; this rate of variation is:

    [13]

    where the evaluation is made at k1 = k because A(k1) is centered there. This result shows that the

    position x where the phase changes slowly, the position where is appreciable, moves with time at a speed called the group velocity:

    The group velocity therefore depends upon the dispersion relation connecting and k. For example, in quantum mechanics the energy of a particle represented as a wave packet is E = = (k)2/(2m). Consequently, for that wave situation, the group velocity is

  • showing that the velocity of a localized particle in quantum mechanics is its group velocity.[13]

    Because the group velocity varies with k, the shape of the wave packet broadens with time, and

    the particle becomes less localized.[16]

    In other words, the velocity of the constituent waves of the

    wave packet travel at a rate that varies with their wavelength, so some move faster than others,

    and they cannot maintain the same interference pattern as the wave propagates.

    Sinusoidal waves

    Sinusoidal waves correspond to simple harmonic motion.

    Mathematically, the most basic wave is the (spatially) one-dimensional sine wave (or harmonic

    wave or sinusoid) with an amplitude described by the equation:

    where

    is the maximum amplitude of the wave, maximum distance from the highest point of

    the disturbance in the medium (the crest) to the equilibrium point during one wave cycle.

    In the illustration to the right, this is the maximum vertical distance between the baseline

    and the wave.

    is the space coordinate

    is the time coordinate

    is the wavenumber

    is the angular frequency

    is the phase constant.

    The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a

    wave on a string) have an amplitude expressed as a distance (e.g., meters), longitudinal

    mechanical waves (e.g., sound waves) use units of pressure (e.g., pascals), and electromagnetic

    waves (a form of transverse vacuum wave) express the amplitude in terms of its electric field

    (e.g., volts/meter).

    The wavelength is the distance between two sequential crests or troughs (or other equivalent

    points), generally is measured in meters. A wavenumber , the spatial frequency of the wave in

    radians per unit distance (typically per meter), can be associated with the wavelength by the

    relation

  • The period is the time for one complete cycle of an oscillation of a wave. The frequency is

    the number of periods per unit time (per second) and is typically measured in hertz. These are

    related by:

    In other words, the frequency and period of a wave are reciprocals.

    The angular frequency represents the frequency in radians per second. It is related to the

    frequency or period by

    The wavelength of a sinusoidal waveform traveling at constant speed is given by:[17]

    where is called the phase speed (magnitude of the phase velocity) of the wave and is the

    wave's frequency.

    Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an

    ocean wave approaching shore, the incoming wave undulates with a varying local wavelength

    that depends in part on the depth of the sea floor compared to the wave height. The analysis of

    the wave can be based upon comparison of the local wavelength with the local water depth.[18]

    Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant

    systems, in the presence of dispersion the sine wave is the unique shape that will propagate

    unchanged but for phase and amplitude, making it easy to analyze.[19]

    Due to the KramersKronig relations, a linear medium with dispersion also exhibits loss, so the sine wave

    propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon

    the medium.[20]

    The sine function is periodic, so the sine wave or sinusoid has a wavelength in

    space and a period in time.[21][22]

    The sinusoid is defined for all times and distances, whereas in physical situations we usually deal

    with waves that exist for a limited span in space and duration in time. Fortunately, an arbitrary

    wave shape can be decomposed into an infinite set of sinusoidal waves by the use of Fourier

    analysis. As a result, the simple case of a single sinusoidal wave can be applied to more general

    cases.[23][24]

    In particular, many media are linear, or nearly so, so the calculation of arbitrary

    wave behavior can be found by adding up responses to individual sinusoidal waves using the

  • superposition principle to find the solution for a general waveform.[25]

    When a medium is

    nonlinear, the response to complex waves cannot be determined from a sine-wave

    decomposition.

    Plane waves

    Main article: Plane wave

    Standing waves

    Main articles: Standing wave, Acoustic resonance, Helmholtz resonator and Organ pipe

    Standing wave in stationary medium. The red dots represent the wave nodes

    A standing wave, also known as a stationary wave, is a wave that remains in a constant position.

    This phenomenon can occur because the medium is moving in the opposite direction to the wave,

    or it can arise in a stationary medium as a result of interference between two waves traveling in

    opposite directions.

    The sum of two counter-propagating waves (of equal amplitude and frequency) creates a

    standing wave. Standing waves commonly arise when a boundary blocks further propagation of

    the wave, thus causing wave reflection, and therefore introducing a counter-propagating wave.

    For example when a violin string is displaced, transverse waves propagate out to where the string

    is held in place at the bridge and the nut, where the waves are reflected back. At the bridge and

    nut, the two opposed waves are in antiphase and cancel each other, producing a node. Halfway

    between two nodes there is an antinode, where the two counter-propagating waves enhance each

    other maximally. There is no net propagation of energy over time.

    One-dimensional standing waves; the fundamental mode and the first 5 overtones.

  • A two-dimensional standing wave on a disk; this is the fundamental mode.

    A standing wave on a disk with two nodal lines crossing at the center; this is an overtone.

    Physical properties

    Light beam exhibiting reflection, refraction, transmission and dispersion when encountering a

    prism

    Waves exhibit common behaviors under a number of standard situations, e. g.

    Transmission and media

    Main articles: Rectilinear propagation, Transmittance and Transmission medium

    Waves normally move in a straight line (i.e. rectilinearly) through a transmission medium. Such

    media can be classified into one or more of the following categories:

    A bounded medium if it is finite in extent, otherwise an unbounded medium

    A linear medium if the amplitudes of different waves at any particular point in the

    medium can be added

    A uniform medium or homogeneous medium if its physical properties are unchanged at

    different locations in space

    An anisotropic medium if one or more of its physical properties differ in one or more

    directions

    An isotropic medium if its physical properties are the same in all directions

    Absorption

    Main articles: Absorption (acoustics) and Absorption (electromagnetic radiation)

  • Absorption of waves mean, if a kind of wave strikes a matter, it will be absorbed by the matter.

    When a wave with that same natural frequency impinges upon an atom, then the electrons of that

    atom will be set into vibrational motion. If a wave of a given frequency strikes a material with

    electrons having the same vibrational frequencies, then those electrons will absorb the energy of

    the wave and transform it into vibrational motion.

    Reflection

    Main article: Reflection (physics)

    When a wave strikes a reflective surface, it changes direction, such that the angle made by the

    incident wave and line normal to the surface equals the angle made by the reflected wave and the

    same normal line.

    Interference

    Main article: Interference (wave propagation)

    Waves that encounter each other combine through superposition to create a new wave called an

    interference pattern. Important interference patterns occur for waves that are in phase.

    Refraction

    Main article: Refraction

    Sinusoidal traveling plane wave entering a region of lower wave velocity at an angle, illustrating

    the decrease in wavelength and change of direction (refraction) that results.

    Refraction is the phenomenon of a wave changing its speed. Mathematically, this means that the

    size of the phase velocity changes. Typically, refraction occurs when a wave passes from one

    medium into another. The amount by which a wave is refracted by a material is given by the

    refractive index of the material. The directions of incidence and refraction are related to the

    refractive indices of the two materials by Snell's law.

    Diffraction

    Main article: Diffraction

  • A wave exhibits diffraction when it encounters an obstacle that bends the wave or when it

    spreads after emerging from an opening. Diffraction effects are more pronounced when the size

    of the obstacle or opening is comparable to the wavelength of the wave.

    Polarization

    Main article: Polarization (waves)

    A wave is polarized if it oscillates in one direction or plane. A wave can be polarized by the use

    of a polarizing filter. The polarization of a transverse wave describes the direction of oscillation

    in the plane perpendicular to the direction of travel.

    Longitudinal waves such as sound waves do not exhibit polarization. For these waves the

    direction of oscillation is along the direction of travel.

    Dispersion

    Schematic of light being dispersed by a prism. Click to see animation.

    Main articles: Dispersion (optics) and Dispersion (water waves)

    A wave undergoes dispersion when either the phase velocity or the group velocity depends on

    the wave frequency. Dispersion is most easily seen by letting white light pass through a prism,

    the result of which is to produce the spectrum of colours of the rainbow. Isaac Newton

    performed experiments with light and prisms, presenting his findings in the Opticks (1704) that

    white light consists of several colours and that these colours cannot be decomposed any

    further.[26]

    Mechanical waves

    Main article: Mechanical wave

    Waves on strings

    Main article: Vibrating string

  • The speed of a transverse wave traveling along a vibrating string ( v ) is directly proportional to

    the square root of the tension of the string ( T ) over the linear mass density ( ):

    where the linear density is the mass per unit length of the string.

    Acoustic waves

    Acoustic or sound waves travel at speed given by

    or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed

    of sound).

    Water waves

    Main article: Water waves

    Ripples on the surface of a pond are actually a combination of transverse and longitudinal

    waves; therefore, the points on the surface follow orbital paths.

    Sounda mechanical wave that propagates through gases, liquids, solids and plasmas; Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;

    Ocean surface waves, which are perturbations that propagate through water.

    Seismic waves

    Main article: Seismic waves

    Shock waves

  • Main article: Shock wave

    See also: Sonic boom and Cherenkov radiation

    Other

    Waves of traffic, that is, propagation of different densities of motor vehicles, and so forth,

    which can be modeled as kinematic waves[27]

    Metachronal wave refers to the appearance of a traveling wave produced by coordinated

    sequential actions.

    It is worth noting that the mass-energy equivalence equation can be solved for this form:

    .

    Electromagnetic waves

    Main articles: Electromagnetic radiation and Electromagnetic spectrum

    (radio, micro, infrared, visible, uv)

    An electromagnetic wave consists of two waves that are oscillations of the electric and magnetic

    fields. An electromagnetic wave travels in a direction that is at right angles to the oscillation

    direction of both fields. In the 19th century, James Clerk Maxwell showed that, in vacuum, the

    electric and magnetic fields satisfy the wave equation both with speed equal to that of the speed

    of light. From this emerged the idea that light is an electromagnetic wave. Electromagnetic

    waves can have different frequencies (and thus wavelengths), giving rise to various types of

    radiation such as radio waves, microwaves, infrared, visible light, ultraviolet and X-rays.

  • Quantum mechanical waves

    Main article: Schrdinger equation

    See also: Wave function

    The Schrdinger equation describes the wave-like behavior of particles in quantum mechanics.

    Solutions of this equation are wave functions which can be used to describe the probability

    density of a particle.

    A propagating wave packet; in general, the envelope of the wave packet moves at a different

    speed than the constituent waves.[28]

    de Broglie waves

    Main articles: Wave packet and Matter wave

    Louis de Broglie postulated that all particles with momentum have a wavelength

    where h is Planck's constant, and p is the magnitude of the momentum of the particle. This

    hypothesis was at the basis of quantum mechanics. Nowadays, this wavelength is called the de

    Broglie wavelength. For example, the electrons in a CRT display have a de Broglie wavelength

    of about 1013

    m.

    A wave representing such a particle traveling in the k-direction is expressed by the wave function

    as follows:

    where the wavelength is determined by the wave vector k as:

    and the momentum by:

  • However, a wave like this with definite wavelength is not localized in space, and so cannot

    represent a particle localized in space. To localize a particle, de Broglie proposed a superposition

    of different wavelengths ranging around a central value in a wave packet,[29]

    a waveform often

    used in quantum mechanics to describe the wave function of a particle. In a wave packet, the

    wavelength of the particle is not precise, and the local wavelength deviates on either side of the

    main wavelength value.

    In representing the wave function of a localized particle, the wave packet is often taken to have a

    Gaussian shape and is called a Gaussian wave packet.[30]

    Gaussian wave packets also are used to

    analyze water waves.[31]

    For example, a Gaussian wavefunction might take the form:[32]

    at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as

    0 = 2 / k0. It is well known from the theory of Fourier analysis,[33]

    or from the Heisenberg

    uncertainty principle (in the case of quantum mechanics) that a narrow range of wavelengths is

    necessary to produce a localized wave packet, and the more localized the envelope, the larger the

    spread in required wavelengths. The Fourier transform of a Gaussian is itself a Gaussian.[34]

    Given the Gaussian:

    the Fourier transform is:

    The Gaussian in space therefore is made up of waves:

    that is, a number of waves of wavelengths such that k = 2 .

    The parameter decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows a spread in wave vector k determined by 1/. That is, the smaller the extent in space, the larger the extent in k, and hence in = 2/k.

  • Animation showing the effect of a cross-polarized gravitational wave on a ring of test particles

    Gravity waves

    Gravity waves are waves generated in a fluid medium or at the interface between two media

    when the force of gravity or buoyancy tries to restore equilibrium. A ripple on a pond is one

    example.

    Gravitational waves

    Main article: Gravitational wave

    Researchers believe that gravitational waves also travel through space, although gravitational

    waves have never been directly detected. Gravitational waves are disturbances in the curvature of

    spacetime, predicted by Einstein's theory of general relativity.

    WKB method

    Main article: WKB method

    See also: Slowly varying envelope approximation

    In a nonuniform medium, in which the wavenumber k can depend on the location as well as the

    frequency, the phase term kx is typically replaced by the integral of k(x)dx, according to the

    WKB method. Such nonuniform traveling waves are common in many physical problems,

    including the mechanics of the cochlea and waves on hanging ropes.

    See also

    Audience wave

    Beat waves

    Capillary waves

    Cymatics

  • Doppler effect

    Envelope detector

    Group velocity

    Harmonic

    Inertial wave

    Index of wave articles

    List of waves named after people

    Ocean surface wave

    Phase velocity

    Reaction-diffusion equation

    Resonance

    Ripple tank

    Rogue wave

    Shallow water equations

    Shive wave machine

    Sound wave

    Standing wave

    Transmission medium

    Wave turbulence

    Waves in plasmas

    References

    1.

    Lev A. Ostrovsky & Alexander I. Potapov (2002). Modulated waves: theory and application. Johns Hopkins University Press. ISBN 0-8018-7325-8.

    Michael A. Slawinski (2003). "Wave equations". Seismic waves and rays in elastic media. Elsevier. pp. 131 ff. ISBN 0-08-043930-6.

    Karl F Graaf (1991). Wave motion in elastic solids (Reprint of Oxford 1975 ed.). Dover. pp. 1314. ISBN 978-0-486-66745-4.

    For an example derivation, see the steps leading up to eq. (17) in Francis Redfern. "Kinematic Derivation of the Wave Equation". Physics Journal.

    Jalal M. Ihsan Shatah, Michael Struwe (2000). "The linear wave equation". Geometric wave equations. American Mathematical Society Bookstore. pp. 37 ff. ISBN 0-8218-2749-9.

    Louis Lyons (1998). All you wanted to know about mathematics but were afraid to ask. Cambridge University Press. pp. 128 ff. ISBN 0-521-43601-X.

    Alexander McPherson (2009). "Waves and their properties". Introduction to Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0-470-18590-2.

    Christian Jirauschek (2005). FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection. Cuvillier Verlag. p. 9. ISBN 3-86537-419-0.

    Fritz Kurt Kneubhl (1997). Oscillations and waves. Springer. p. 365. ISBN 3-540-62001-X.

    Mark Lundstrom (2000). Fundamentals of carrier transport. Cambridge University Press. p. 33. ISBN 0-521-63134-3.

  • Chin-Lin Chen (2006). "13.7.3 Pulse envelope in nondispersive media". Foundations for guided-wave optics. Wiley. p. 363. ISBN 0-471-75687-3.

    Stefano Longhi, Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals". In Hugo E. Hernndez-Figueroa, Michel Zamboni-Rached, Erasmo Recami. Localized

    Waves. Wiley-Interscience. p. 329. ISBN 0-470-10885-1.

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    See, for example, Eq. 2(a) in Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics: An introduction (2nd ed.). Springer. pp. 6061. ISBN 3-540-67458-6.

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    See Eq. 5.10 and discussion in A. G. G. M. Tielens (2005). The physics and chemistry of the interstellar medium. Cambridge University Press. pp. 119 ff. ISBN 0-521-82634-9.; Eq. 6.36 and

    associated discussion in Otfried Madelung (1996). Introduction to solid-state theory (3rd ed.).

    Springer. pp. 261 ff. ISBN 3-540-60443-X.; and Eq. 3.5 in F Mainardi (1996). "Transient waves

    in linear viscoelastic media". In Ardshir Guran, A. Bostrom, Herbert berall, O. Leroy.

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    Seth Stein, Michael E. Wysession (2003). op. cit.. p. 32. ISBN 0-86542-078-5.

    Kimball A. Milton, Julian Seymour Schwinger (2006). Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators. Springer. p. 16. ISBN 3-540-29304-3. Thus,

    an arbitrary function f(r, t) can be synthesized by a proper superposition of the functions exp[i

    (krt)]...

    Raymond A. Serway and John W. Jewett (2005). "14.1 The Principle of Superposition". Principles of physics (4th ed.). Cengage Learning. p. 433. ISBN 0-534-49143-X.

    Newton, Isaac (1704). "Prop VII Theor V". Opticks: Or, A treatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of

    Curvilinear Figures 1. London. p. 118. All the Colours in the Universe which are made by

    Light... are either the Colours of homogeneal Lights, or compounded of these...

    M. J. Lighthill; G. B. Whitham (1955). "On kinematic waves. II. A theory of traffic flow on long crowded roads". Proceedings of the Royal Society of London. Series A 229: 281345. Bibcode:1955RSPSA.229..281L. doi:10.1098/rspa.1955.0088. And: P. I. Richards (1956).

    "Shockwaves on the highway". Operations Research 4 (1): 4251. doi:10.1287/opre.4.1.42.

  • A. T. Fromhold (1991). "Wave packet solutions". Quantum Mechanics for Applied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff.

    ISBN 0-486-66741-3. (p. 61) ...the individual waves move more slowly than the packet and

    therefore pass back through the packet as it advances

    Ming Chiang Li (1980). "Electron Interference". In L. Marton & Claire Marton. Advances in Electronics and Electron Physics 53. Academic Press. p. 271. ISBN 0-12-014653-3.

    See for example Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (2 ed.). Springer. p. 60. ISBN 3-540-67458-6. and John Joseph Gilman (2003). Electronic basis of the

    strength of materials. Cambridge University Press. p. 57. ISBN 0-521-62005-8.,Donald D. Fitts

    (1999). Principles of quantum mechanics. Cambridge University Press. p. 17. ISBN 0-521-

    65841-1..

    Chiang C. Mei (1989). The applied dynamics of ocean surface waves (2nd ed.). World Scientific. p. 47. ISBN 9971-5-0789-7.

    Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (2nd ed.). Springer. p. 60. ISBN 3-540-67458-6.

    Siegmund Brandt, Hans Dieter Dahmen (2001). The picture book of quantum mechanics (3rd ed.). Springer. p. 23. ISBN 0-387-95141-5.

    34. Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 677. ISBN 0-521-59827-3.

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